![]() |
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > hdmap1l6i | Structured version Visualization version GIF version |
Description: Lemmma for hdmap1l6 37842. Eliminate auxiliary vector 𝑤. (Contributed by NM, 1-May-2015.) |
Ref | Expression |
---|---|
hdmap1l6.h | ⊢ 𝐻 = (LHyp‘𝐾) |
hdmap1l6.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
hdmap1l6.v | ⊢ 𝑉 = (Base‘𝑈) |
hdmap1l6.p | ⊢ + = (+g‘𝑈) |
hdmap1l6.s | ⊢ − = (-g‘𝑈) |
hdmap1l6c.o | ⊢ 0 = (0g‘𝑈) |
hdmap1l6.n | ⊢ 𝑁 = (LSpan‘𝑈) |
hdmap1l6.c | ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) |
hdmap1l6.d | ⊢ 𝐷 = (Base‘𝐶) |
hdmap1l6.a | ⊢ ✚ = (+g‘𝐶) |
hdmap1l6.r | ⊢ 𝑅 = (-g‘𝐶) |
hdmap1l6.q | ⊢ 𝑄 = (0g‘𝐶) |
hdmap1l6.l | ⊢ 𝐿 = (LSpan‘𝐶) |
hdmap1l6.m | ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) |
hdmap1l6.i | ⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊) |
hdmap1l6.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
hdmap1l6.f | ⊢ (𝜑 → 𝐹 ∈ 𝐷) |
hdmap1l6cl.x | ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) |
hdmap1l6.mn | ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹})) |
hdmap1l6i.xn | ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍})) |
hdmap1l6i.y | ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) |
hdmap1l6i.z | ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 })) |
hdmap1l6i.yz | ⊢ (𝜑 → (𝑁‘{𝑌}) = (𝑁‘{𝑍})) |
Ref | Expression |
---|---|
hdmap1l6i | ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, (𝑌 + 𝑍)〉) = ((𝐼‘〈𝑋, 𝐹, 𝑌〉) ✚ (𝐼‘〈𝑋, 𝐹, 𝑍〉))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hdmap1l6.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | hdmap1l6.u | . . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
3 | hdmap1l6.v | . . 3 ⊢ 𝑉 = (Base‘𝑈) | |
4 | hdmap1l6.n | . . 3 ⊢ 𝑁 = (LSpan‘𝑈) | |
5 | hdmap1l6.k | . . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
6 | hdmap1l6cl.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) | |
7 | 6 | eldifad 3781 | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
8 | hdmap1l6i.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) | |
9 | 8 | eldifad 3781 | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
10 | 1, 2, 3, 4, 5, 7, 9 | dvh3dim 37467 | . 2 ⊢ (𝜑 → ∃𝑤 ∈ 𝑉 ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) |
11 | hdmap1l6.p | . . . 4 ⊢ + = (+g‘𝑈) | |
12 | hdmap1l6.s | . . . 4 ⊢ − = (-g‘𝑈) | |
13 | hdmap1l6c.o | . . . 4 ⊢ 0 = (0g‘𝑈) | |
14 | hdmap1l6.c | . . . 4 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) | |
15 | hdmap1l6.d | . . . 4 ⊢ 𝐷 = (Base‘𝐶) | |
16 | hdmap1l6.a | . . . 4 ⊢ ✚ = (+g‘𝐶) | |
17 | hdmap1l6.r | . . . 4 ⊢ 𝑅 = (-g‘𝐶) | |
18 | hdmap1l6.q | . . . 4 ⊢ 𝑄 = (0g‘𝐶) | |
19 | hdmap1l6.l | . . . 4 ⊢ 𝐿 = (LSpan‘𝐶) | |
20 | hdmap1l6.m | . . . 4 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) | |
21 | hdmap1l6.i | . . . 4 ⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊) | |
22 | 5 | 3ad2ant1 1164 | . . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
23 | hdmap1l6.f | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ 𝐷) | |
24 | 23 | 3ad2ant1 1164 | . . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → 𝐹 ∈ 𝐷) |
25 | 6 | 3ad2ant1 1164 | . . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → 𝑋 ∈ (𝑉 ∖ { 0 })) |
26 | hdmap1l6.mn | . . . . 5 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹})) | |
27 | 26 | 3ad2ant1 1164 | . . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹})) |
28 | hdmap1l6i.xn | . . . . 5 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍})) | |
29 | 28 | 3ad2ant1 1164 | . . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍})) |
30 | hdmap1l6i.yz | . . . . 5 ⊢ (𝜑 → (𝑁‘{𝑌}) = (𝑁‘{𝑍})) | |
31 | 30 | 3ad2ant1 1164 | . . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → (𝑁‘{𝑌}) = (𝑁‘{𝑍})) |
32 | 8 | 3ad2ant1 1164 | . . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → 𝑌 ∈ (𝑉 ∖ { 0 })) |
33 | hdmap1l6i.z | . . . . 5 ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 })) | |
34 | 33 | 3ad2ant1 1164 | . . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → 𝑍 ∈ (𝑉 ∖ { 0 })) |
35 | eqid 2799 | . . . . 5 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈) | |
36 | 1, 2, 5 | dvhlmod 37131 | . . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LMod) |
37 | 36 | 3ad2ant1 1164 | . . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → 𝑈 ∈ LMod) |
38 | 3, 35, 4, 36, 7, 9 | lspprcl 19299 | . . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ∈ (LSubSp‘𝑈)) |
39 | 38 | 3ad2ant1 1164 | . . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → (𝑁‘{𝑋, 𝑌}) ∈ (LSubSp‘𝑈)) |
40 | simp2 1168 | . . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → 𝑤 ∈ 𝑉) | |
41 | simp3 1169 | . . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) | |
42 | 13, 35, 37, 39, 40, 41 | lssneln0 19271 | . . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → 𝑤 ∈ (𝑉 ∖ { 0 })) |
43 | 1, 2, 3, 11, 12, 13, 4, 14, 15, 16, 17, 18, 19, 20, 21, 22, 24, 25, 27, 29, 31, 32, 34, 42, 41 | hdmap1l6h 37838 | . . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → (𝐼‘〈𝑋, 𝐹, (𝑌 + 𝑍)〉) = ((𝐼‘〈𝑋, 𝐹, 𝑌〉) ✚ (𝐼‘〈𝑋, 𝐹, 𝑍〉))) |
44 | 43 | rexlimdv3a 3214 | . 2 ⊢ (𝜑 → (∃𝑤 ∈ 𝑉 ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}) → (𝐼‘〈𝑋, 𝐹, (𝑌 + 𝑍)〉) = ((𝐼‘〈𝑋, 𝐹, 𝑌〉) ✚ (𝐼‘〈𝑋, 𝐹, 𝑍〉)))) |
45 | 10, 44 | mpd 15 | 1 ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, (𝑌 + 𝑍)〉) = ((𝐼‘〈𝑋, 𝐹, 𝑌〉) ✚ (𝐼‘〈𝑋, 𝐹, 𝑍〉))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 385 ∧ w3a 1108 = wceq 1653 ∈ wcel 2157 ∃wrex 3090 ∖ cdif 3766 {csn 4368 {cpr 4370 〈cotp 4376 ‘cfv 6101 (class class class)co 6878 Basecbs 16184 +gcplusg 16267 0gc0g 16415 -gcsg 17740 LModclmod 19181 LSubSpclss 19250 LSpanclspn 19292 HLchlt 35371 LHypclh 36005 DVecHcdvh 37099 LCDualclcd 37607 mapdcmpd 37645 HDMap1chdma1 37812 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-rep 4964 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-cnex 10280 ax-resscn 10281 ax-1cn 10282 ax-icn 10283 ax-addcl 10284 ax-addrcl 10285 ax-mulcl 10286 ax-mulrcl 10287 ax-mulcom 10288 ax-addass 10289 ax-mulass 10290 ax-distr 10291 ax-i2m1 10292 ax-1ne0 10293 ax-1rid 10294 ax-rnegex 10295 ax-rrecex 10296 ax-cnre 10297 ax-pre-lttri 10298 ax-pre-lttrn 10299 ax-pre-ltadd 10300 ax-pre-mulgt0 10301 ax-riotaBAD 34974 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-fal 1667 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-pss 3785 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-tp 4373 df-op 4375 df-ot 4377 df-uni 4629 df-int 4668 df-iun 4712 df-iin 4713 df-br 4844 df-opab 4906 df-mpt 4923 df-tr 4946 df-id 5220 df-eprel 5225 df-po 5233 df-so 5234 df-fr 5271 df-we 5273 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-pred 5898 df-ord 5944 df-on 5945 df-lim 5946 df-suc 5947 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-riota 6839 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-of 7131 df-om 7300 df-1st 7401 df-2nd 7402 df-tpos 7590 df-undef 7637 df-wrecs 7645 df-recs 7707 df-rdg 7745 df-1o 7799 df-oadd 7803 df-er 7982 df-map 8097 df-en 8196 df-dom 8197 df-sdom 8198 df-fin 8199 df-pnf 10365 df-mnf 10366 df-xr 10367 df-ltxr 10368 df-le 10369 df-sub 10558 df-neg 10559 df-nn 11313 df-2 11376 df-3 11377 df-4 11378 df-5 11379 df-6 11380 df-n0 11581 df-z 11667 df-uz 11931 df-fz 12581 df-struct 16186 df-ndx 16187 df-slot 16188 df-base 16190 df-sets 16191 df-ress 16192 df-plusg 16280 df-mulr 16281 df-sca 16283 df-vsca 16284 df-0g 16417 df-mre 16561 df-mrc 16562 df-acs 16564 df-proset 17243 df-poset 17261 df-plt 17273 df-lub 17289 df-glb 17290 df-join 17291 df-meet 17292 df-p0 17354 df-p1 17355 df-lat 17361 df-clat 17423 df-mgm 17557 df-sgrp 17599 df-mnd 17610 df-submnd 17651 df-grp 17741 df-minusg 17742 df-sbg 17743 df-subg 17904 df-cntz 18062 df-oppg 18088 df-lsm 18364 df-cmn 18510 df-abl 18511 df-mgp 18806 df-ur 18818 df-ring 18865 df-oppr 18939 df-dvdsr 18957 df-unit 18958 df-invr 18988 df-dvr 18999 df-drng 19067 df-lmod 19183 df-lss 19251 df-lsp 19293 df-lvec 19424 df-lsatoms 34997 df-lshyp 34998 df-lcv 35040 df-lfl 35079 df-lkr 35107 df-ldual 35145 df-oposet 35197 df-ol 35199 df-oml 35200 df-covers 35287 df-ats 35288 df-atl 35319 df-cvlat 35343 df-hlat 35372 df-llines 35519 df-lplanes 35520 df-lvols 35521 df-lines 35522 df-psubsp 35524 df-pmap 35525 df-padd 35817 df-lhyp 36009 df-laut 36010 df-ldil 36125 df-ltrn 36126 df-trl 36180 df-tgrp 36764 df-tendo 36776 df-edring 36778 df-dveca 37024 df-disoa 37050 df-dvech 37100 df-dib 37160 df-dic 37194 df-dih 37250 df-doch 37369 df-djh 37416 df-lcdual 37608 df-mapd 37646 df-hdmap1 37814 |
This theorem is referenced by: hdmap1l6j 37840 |
Copyright terms: Public domain | W3C validator |